example of telescoping sum

Some trigonometric sums, as ∑k=1ncos⁡k⁢α and ∑k=1nsin⁡k⁢α, may be telescoped if the terms are first edited by a suitable goniometric formula (http://planetmath.org/GoniometricFormulae) (‘‘product formula’’). E.g. we may write:

∑k=1ncos⁡k⁢α=1sin⁡α2⁢∑k=1ncos⁡k⁢α⁢sin⁡α2

The product formula cos⁡x⁢sin⁡y=12⁢[sin⁡(x+y)-sin⁡(x-y)] alters this to